Theorem axext4 2439

Description: A bidirectional version of Extensionality. Although this theorem "looks" like it is just a definition of equality, it requires the Axiom of Extensionality for its proof under our axiomatization. See the comments for ax-ext 2435 and df-cleq 2449. (Contributed by NM, 14-Nov-2008.)

Assertion

Ref	Expression
axext4

Distinct variable groups:   ,   ,

Proof of Theorem axext4

Step	Hyp	Ref	Expression
1		elequ2 1823	. . 3
2	1	alrimiv 1719	. 2
3		axext3 2437	. 2
4	2, 3	impbii 188	1

Syntax hints:  <->wb 184  A.wal 1393

This theorem is referenced by:  axc11next  31313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-ex 1613